Schatten classes of integration operators on Dirichlet spaces

We address the problem of determining membership in Schatten-Von Neumann ideals Sp of integration operators (Tgf)(z) = ∫0z = ∫0zf(ξ)g′(ξ)dξ acting on Dirichlet type spaces. We also study this problem for multiplication, Hankel and Toeplitz operators. In particular, we provide an extension of Luecking's result on Toeplitz operators [10, p. 347].

For 0 < p < ∞, we also write A p α for the weighted Bergman space of those g ∈ H (D) such that g p A p α = D |g(z)| p dA α (z) < ∞. Here, we focus our attention on the study of the integration operator T g and the multiplication operator M g defined by where g is an analytic function on D.
The bilinear operator ( f, g) → fg ′ was introduced by A. Calderón in harmonic analysis in the 1960's for his research on commutators of singular integral operators [8] (see also [25, p. 1136]). Later, it and different variations going by the name of "paraproducts" have been extensively studied, and the operator has become a fundamental tool in harmonic analysis. Pommerenke was probably one of the first to consider the operator T g [17]. In recent years, following the pioneering works of Aleman and Siskakis [4,5], the study of the operator T g on several spaces of analytic functions has attracted much attention; see [2,3,14,16,22,23].
Our main goal is to determine conditions under which the integration operator T g : D α → D α belongs to the Schatten-Von Neumann ideal S p . If α > 1, D α is nothing other than A 2 α−2 and D 1 is the classical Hardy space H 2 . Thus, for p > 1, T g ∈ S p (D α ) if and only if g ∈ B p , and if 0 < p ≤ 1, then T g ∈ S p (D α ) if and only if g is constant; see [4,5]. We recall that for p > 1, the Besov space B p is the space of analytic functions g in D such that D |g ′ (z)| p (1 − |z| 2 ) p dλ(z) < ∞, where dλ(z) = dA(z)/(1 − |z| 2 ) 2 is the hyperbolic measure on D.
The following result is implicit in the literature (see [27]) and can be proved using the theory of Toeplitz operators; see Section 5.
To the best of our knowledge, describing those g ∈ H (D) for which T g ∈ S p (D α ) when 0 < α < 1 and p(1 − α) ≥ 2 is an open problem. In order to study this problem, we introduce, for 0 ≤ α < ∞ and 1 < p < ∞, the space X p α which consists of those g ∈ H (D) such that The following result describes those g ∈ H (D) such that T g ∈ S p (D α ) in the range p > 1 and p(1 − α) < 4. Theorem 1. Let 0 < α < 1, g ∈ H (D), and p > 1 with p(1 − α) < 4. Then T g ∈ S p (D α ) if and only if g ∈ X p α .
We now deal with the case of the classical Dirichlet space D. This situation seems to be much more difficult. First of all, it is easy (and well known how) to determine when an operator T g belongs to the Hilbert-Schmidt class S 2 (D). Indeed, for every orthonormal basis {e n } of the Dirichlet space, (1.2) see Section 2. Therefore, the integration operator T g belongs to S 2 (D) if and only if the last integral in (1.2) is finite. We denote the class of functions g ∈ H (D) satisfying this condition by DL. Theorem A suggests that if 1 < p < 2, the condition g ∈ B p describes the symbols g such that T g is in S p (D). However, for p < 2, every operator on S p must be Hilbert-Schmidt. Thus, clearly the condition g ∈ DL is necessary for T g ∈ S p (D). An easy calculation then shows that the function g(z) = log log e/(1 − z) belongs to B p for all p > 1, but g / ∈ DL. Thus, the condition g ∈ B p is not sufficient for T g to belong to S p (D).
On the other hand, as in the weighted case, there are no trace class integration operators in the Dirichlet space unless g is constant.
Theorem 2. Let 0 < p ≤ 1 and g ∈ H (D). Then T g ∈ S p (D) if and only if g is constant.
For the case 1 < p < 2, we have a necessary condition and a different sufficient condition. It turns out that these conditions are sharp in a certain sense. Before showing this, we consider the space B p,log γ for p > 1 and γ > 0. This space consists of those functions g analytic on D and such that It turns out that with the monomials as symbols, the correct behavior of T g S p is given by B p or X p 0 , while with functions of the type g a (z) = (1 −āz) −γ as symbols, the correct behavior is given by the B p,log p/2 condition; see Lemmas 4.1 and 4.2. The case p > 2 seems to be a mystery. [28, p. 94]. Thus, a necessary condition much better than B p = D p p−2 cannot be expected for a condition involving the integrability of g ′ . Moreover, a sufficient condition must be stronger than g ∈ D p (p−2)/2 . We discuss this case further in Section 4.
The paper is organized as follows. In Section 2, we introduce several preliminary general results related to Schatten classes of operators on Dirichlet spaces. Section 3 is devoted to the proof of Theorem 1. There, the identity is proved directly; see Proposition 3.1(iv). This identity, together with Theorem 1, gives a proof of Theorem A which does not rely on the theory of Toeplitz operators. It is worth mentioning that the Besov space B p has several different characterizations ((1.3) being a new one), each of which is the appropriate tool for analyzing certain situations; see, e.g. [1], [7], or [29]). In Section 4, we prove Theorem 2 and Theorem 3. Also, using particular classes of test functions, we show that these results are sharp in a certain sense.
Finally, Section 5 is devoted to a study of the relationship between the integration operator T g and other classical operators acting on weighted Dirichlet spaces, such as Toeplitz operators, multiplication operators, or big and small Hankel operators. A similar relationship also exists in other contexts; see [18]. Indeed, the same techniques used in the proof of Theorem 1 can be used to obtain an extension of the useful result of Luecking on Toeplitz operators; see [10, p. 347].
Throughout the paper, C denotes a positive absolute constant whose value may be different at different occurrences. We define A 1 and A 2 to be comparable and write A 1 ≍ A 2 if there exist positive constants c 1 and c 2 such that A 1 ≤ c 1 A 2 and A 2 ≤ c 2 A 1 .

Preliminary results
Let H and K be separable Hilbert spaces. Given 0 < p < ∞, let S p (H, K ) denote the Schatten p-class of operators from H to K . If H = K we simply write S p (H ). The class S p (H, K ) consists of those compact operators T from H to K whose sequence of singular numbers λ n belongs to the p-summable sequence space ℓ p . We recall that a singular number of a compact operator T is the square root of an eigenvalue of the positive operator T * T , where T * denotes the Hilbert adjoint of T . We remind the reader that T ∈ S p (H ) if and only if T * T ∈ S p/2 (H ). Also, the compact operator T admits a decomposition of the form T = n λ n ·, e n H σ n , where {λ n } are the singular numbers of T , {e n } is an orthonormal set in H , and {σ n } is an orthonormal set in K .
For p ≥ 1, the class S p (H, K ) equipped with the norm T S p = n |λ n | p 1/p is a Banach space, while for 0 < p < 1, We refer to [21] or [30, Chapter 1] for a brief account of the theory of Schatten p-classes.
We denote by H a Hilbert space of analytic functions in D with reproducing kernels K z . Usually, the reproducing kernel functions carry a large amount of information about relevant properties such as boundedness, compactness, membership in Schatten p-classes, etc. of an operator T on H . It is known that if {e n } is an orthonormal basis of a Hilbert space H of analytic functions in D with reproducing kernel K z , then for all z and ζ in D; see, e.g., [30,Theorem 4.19]. We introduce the derivative J z of K z respect to z In order to avoid confusion, when dealing with reproducing kernels of either D α or A 2 α , we write B α z for the reproducing kernel of the weighted Bergman space A 2 α at the point z and let b α α be its normalization. It is well known (see [30,Corollary 4.20]) that The reproducing kernel function for the Dirichlet type space D α is denoted by K α z , and k α z denotes the corresponding normalized reproducing kernel. Since f ∈ D α if and only if f ′ ∈ A 2 α , the reproducing formula for the Bergman space A 2 α (see [30,Proposition 4.23]) implies the following expression for the reproducing kernel of D α (see [7] or [27]): In particular, for α = 0, K D z (w) : = K 0 z (w) = 1 + log 1/(1 −zw). Also, it is easy to see that While the next two results are certainly well known (see [9] or [24] for similar results), we offer a proof for the reader's convenience.

Proposition 2.1. Let H be a separable Hilbert space and T :
Proof. Since T is compact, it admits the decomposition where {λ n } are the singular values of T , {e n } is an orthonormal set in A 2 α , and { f n } is an orthonormal set in H . Thus TB α z = n λ n e n (z) f n , and therefore TB α If 0 < p ≤ 2, a similar argument, using Hölder's inequality with exponent 2/p ≥ 1, (2.3) and (2.4), gives The corresponding analogue of Proposition 2.1 for the Dirichlet type spaces D α uses the functions j α

Proposition 2.2. Let H be a separable Hilbert space and T
Proof. Since T is compact, it admits the decomposition where {λ n } are the singular values of T , {e n } is an orthonormal set in D α , and { f n } is an orthonormal set in H . It follows from (2.5) that J α z (0) = 0. Equation (2.4) then yields For p ≥ 2, Hölder's inequality, (2.8), (2.3), and (2.7) yield If 0 < p ≤ 2, then . For the term (I ), observe that |λ n | ≤ T , and therefore For the term (II ), because of Hölder's inequality, (2.3), and (2.8). Putting the estimates obtained for (I ) and (II ) into (2.9) gives (ii).
We also need the following result.
Proof. Let {e n } be an orthonormal basis of D α . Equations (2.2) and (2.5) give This gives the result for p = 1.
We also use the following integral estimate (see [30]) several times. This estimate has become indispensable in this area of analysis.
The following useful inequality is from [13]; it can be thought as a generalization of the preceding result.
For z ∈ D and r > 0, let D(z, r) = {w ∈ D : β(z, w) < r} denote the hyperbolic disk with center z and radius r. Here, β(z, w) is the Bergman or hyperbolic metric on D. We also need the concept of an r-lattice in the Bergman metric. Let r > 0. 3 Case 0 < α < 1 Before embarking on the proof of Theorem 1, we prove some preliminary results of independent interest.

A new class of spaces.
In this subsection, we display several nesting properties of X p α and B p spaces. We offer a proof of (1.3) which under the conditions appearing there give an equivalent B p -norm. It is worth mentioning that equivalent and useful B p -norms have been previously introduced for the study of operators on different spaces of analytic functions on D; see [1] and [7], for example. Also, our next result proves that X p This shows that X p α ⊂ D α , proving (i).
(ii) Assume that g ∈ X p α and fix a ∈ D. If w ∈ D(a), then (1 − |w|) ≍ (1 − |a|) and |1 −wz| ≍ |1 −āz| for z ∈ D. Hence This gives and it follows easily that ||g|| q Then, using Hölder's inequality and Lemma B, we obtain Note that the choice of ε gives pα > β, and therefore we can use Lemma B again in order to obtain Now assume that 1 < p ≤ 2 and fix an r-lattice {a n } with associated hyperbolic disks {D n }. Then Now, interchanging the order of summation and integration and using Lemma B, we obtain where the last step follows from Theorem 0 of [5]; see also [29].
3.2 Proof of Theorem 1. Sufficiency for the case 1 < p ≤ 2 and necessity for 2 ≤ p < ∞ are byproducts of the following result, which also gives some information on the case p(1 − α) > 4.
, the result follows directly from Proposition 2.2.
The necessity for 1 < p < 2 follows from the next proposition and part (iv) of Proposition 3.1.
Proof. Let 1 ≤ p < 2, and assume that T g ∈ S p (D α ). Then the positive operator T * g T g belongs to S p/2 (D α ). Without loss of generality, we may assume that g ′ = 0. Suppose that T * g T g f = n λ n f, e n e n is the canonical decomposition of T * g T g . Then {e n } is not only an orthonormal set, but is also an orthonormal basis. Indeed, if there exists a unit vector e ∈ D α such that e ⊥ e n for all n ≥ 1, then because T * g T g is a linear combination of the vectors e n . This gives g ′ ≡ 0, a contradiction.
Since {e n } is an orthonormal basis of D α , it follows from Lemma 2.3 that This completes the proof of (i).
which implies that g is constant.
The remaining part of the proof Theorem 1 is more involved. We split it into two cases. with Since g ∈ X p α ⊂ D α by Proposition 3.1 and |e n (0)| ≤ 1, we clearly have In order to deal with the term I 2 , note first that e 2 n ∈ D 1+2α because for all f ∈ D α , So, from the reproducing formula for D 1+2α , we deduce , Fubini's theorem and Hölder's inequality yield If p = 4, it follows from (2.3) and the fact that K α w 2 α . Now, observe that Hölder's inequality with exponent 4/p > 1 and (2.3) yield, for 2 < p < 4, Combining the estimates for I 2 and I 1 yields n T g e n p D α ≤ C < ∞, since g ∈ X p α . Thus, by [30, Theorem 1.33], the operator T g belongs to S p (D α ).

The open case.
As pertains to the open case p(1 − α) ≥ 4, we state the following result, which can be proved following the lines of the proof of Theorem 1 (case p > 4). We thus omit the proof. Proposition 3.4. Let 0 < α, p ≥ 2 , and g ∈ H (D). If g ∈ X p α−ε for some ε ∈ (0, α), then T g ∈ S p (D α ).
Since X p α−ε X p α if (1 − α)p ≥ 2 (see Lemma 4.1 below), Proposition 3.4 gives a sufficient (but not necessary) condition for T g ∈ S p (D α ), (1 − α)p ≥ 2. However, if α > 0 and 1 < p < ∞, techniques developed in our proof of Lemma 4.2 together with Lemma C imply that for all β > 0, In particular, the condition X p α gives the right growth for this family of functions.

Proof of Theorem 3. Part (a) follows from part (i) of Proposition 3.3, and part (c) is deduced in Proposition 3.2.
In order to prove part (b), assume that 1 < p < 2. Then, for all orthonormal sets {e n } of D, Thus, from [30, Theorem 1.27], it follows that T g ∈ S p (D) with T g S p ≤ C g B p log p/2 .

Testing functions for Schatten classes.
Our next goal is a proof that Theorem 3 gives the correct behavior of T g S p , 1 < p < 2, at least for some families of functions. We begin by considering monomials. Lemma 4.1. Assume that 0 ≤ α < 1 and 1 < p < ∞. Let g j (z) = z j , j = 1, 2, 3 . . .. Then Proof. For f (z) = ∞ n =0 a n z n , and g(z) = ∞ n =0 b n z n , consider the inner product on D α given by f, g = ∞ k =0 (k + 1) 1−α a k b k . We note that a k k + j z k+ j = j ∞ n = j a n− j n z n . Now, if σ n (z) = z n /(n + 1) (1−α)/2 and n ∈ N, {σ n } ∞ n =0 is an orthonomal basis of D α ; furthermore, ∞ n = j a n− j n z n = ∞ n = j a n− j (n + 1) ( i.e., the singular values of the integration operator T g j are Consequently, On the other hand, At this point, we invoke [12, Theorem 1] to obtain which together with (4.4) gives the estimate T g j S p (D α ) ≍ ||g j || X p α in (4.1). The second estimate in (4.1) is a straightforward calculation. Now, to prove (4.2), observe that where, in the last step, we have used the fact that is an admissible weight with distortion function equivalent to (1 − r); see [15, p. 11]. Now, bearing in mind the properties of the beta function, we have The equivalence (4.3) can be proved similarly.

This and Lemma B imply
On the other hand, taking 0 < ε < min(1, 2(p−1)/p), and bearing in mind Lemma C, we have An application of Lemma B then gives To prove (4.7), we first estimate the B p,log p/2 -norm of the functions g a (z) = (1 −āz) −γ . Take a ∈ D with |a| ≥ 1/2. Then Since (1 − s) p−2 (log e/(1 − s)) p/2 is an admissible weight, Moreover, which together with (4.8) and (4.9) gives  Bearing in mind that for p > 1 X p α , || · || X p α is a Banach space, we conclude from the Closed Graph Theorem, Lemma 4.1, and Lemma 4.2 that X p 0 B p and is different from B p,log p/2 . In particular, the statement analogous to Proposition 3.1(iv) for α = 0 and 1 < p < 2 does not hold.

Case p > 2.
We collect our results for this range of values of p in the next proposition. Proposition 4.3. Assume that g ∈ H (D) and 2 ≤ p < ∞.
then T g ∈ S p (D).

Relationship to other operators
It should be noticed that the integration operator T g is bounded, compact (in D α ), or belongs to the Schatten class S p (D α ) if and only if the multiplication operator M g ′ : D α → A 2 α is bounded, compact, or belongs to S p , respectively. In this section, we study the relationship between the integration operator T g (equivalently M g ′ ) and other linear operators such as Toeplitz operators, the big and small Hankel operators, or other multiplication operators.

Toeplitz operators.
Recall that given a finite positive Borel measure µ on D, the Toeplitz operator Q µ on D α , α > 0, is defined by Toeplitz operators have been a key tool for finding conditions for membership in S p of many classes of operators, such as composition operators (see [11], [10, Section 7] and [30, Chapter 11]) or integration operators (see [4,5] and [16,Chapter 6]). Indeed, the integration operator T g and the Toeplitz operator Q µ on D α are related via the identity T * g T g = Q µ g , where µ g is the measure defined by dµ g (z) = |g ′ (z)| 2 dA α (z), and one can obtain a proof of Theorem A using the characterization of Schatten class Toeplitz operators obtained by D. Luecking; see (5.1) below. Thus, it is natural to expect that the methods used to find conditions for membership of T g in the Schatten p-class of D α apply also to the Toeplitz operator Q µ on D α for a general measure µ. Before proving this, we recall Luecking's result describing the membership in S p (D α ) of the Toeplitz operator Q µ for all p > 0 with p(1 − α) < 1.
In [10], he showed that for the range of p considered above, Q µ ∈ S p (D α ) if and only if for any r-lattice {a j } with associated hyperbolic disks {D j }, Given a finite positive Borel measure on D, for all −1 < α < ∞ and 0 < p < ∞, we define Here, we are able to obtain a full description of the measures µ for which the Toeplitz operator Q µ belongs to S p (D α ) on the extended range of all p > 0 with p(1 − α) < 2 and 1 < p(2 + α). Since α > 0, we obtain a complete description of the Hilbert-Schmidt Toeplitz operators on D α . Theorem 5.1. Let µ be a finite positive Borel measure on D, α > 0 and p satisfy p > 0, 1 < p(2 + α), and p(1 − α) < 2. Then the Toeplitz operator Q µ belongs to S p (D α ) if and only if X 2p α (µ) < ∞.
Proof. Consider the inclusion operator I µ : D α → L 2 (D, µ). It is easy to check that Q µ = I * µ I µ , and thus Q µ ∈ S p (D α ) if and only if I µ ∈ S 2p . Now, the necessity of X 2p α (µ) < ∞ for p ≥ 1 and the sufficiency for p ≤ 1 follow from Proposition 2.2. Also, repeating the proof of sufficiency in Theorem 1 replacing the measure |g ′ (z)| 2 dA α (z) with the measure dµ, we obtain n I µ e n 2p L 2 (D,µ) ≤ C < ∞ for all orthonormal sets {e n } of D α , provided that p > 1 and p(1 − α) < 2. This proves the sufficiency of X 2p α (µ) < ∞ in that range. Finally, it remains to show the necessity in the case 1/(2 + α) < p < 1. Let {a j } be an r-lattice with associated hyperbolic disks {D j }. Using the fact that |1 −wz| ≍ |1 −ā j z| for w ∈ D j and Lemma B, we deduce Thus, it follows from Luecking's condition (5.1) We note that in [19], one can find a description of the membership of the Toeplitz operator Q µ in S 2k (D α ) for positive integers k in terms of some iterated integrals. [26] and [20], for α ≥ 0, we consider the Sobolev space L 2 α consisting of those differentiable functions u : D → C for which

Big and small Hankel operators. As in
It is clear that D α is a closed subspace of L 2 α . Let P α be the orthogonal projection of L 2 α onto D α . The big Hankel operator H α g : D α → L 2 α and the small Hankel operator h α g : D α → L 2 α are defined respectively by The relation between the big Hankel operator and the multiplication operator M g ′ is clear and well understood. Indeed, in [26,Corollary 1], Z. Wu showed that M g ′ : D α → A 2 α is bounded, compact, or belongs to S p with 1 < p < ∞ if and only if the same is true for the big Hankel operator H α g : D α → L 2 α . However, although Mḡ′ is related to the the small Hankel operator (see (5.8) [26,Theorem 6]. Note that, by our previous observations, we may replace H 0 g by M g ′ or T g .) The main aim of this section is to extend Wu's result on Schatten p-classes for the small Hankel operator to all of D α and to all p, 1 < p < ∞. Before doing so, we recall that and that P α u(w) satisfies (see [20, p. 105]) Lemma 5.2. Let σ > −1, and 2 + σ < b ≤ 4 + 2σ. Then for each a ∈ D and f ∈ H (D), For u ∈ L 2 (D, dA α ), consider the operator Proof. We deal with the case p ≥ 2 first. Note that if f ∈ H ∞ (the algebra of all bounded analytic functions on D, a dense subset of D α ) and u is analytic, then Therefore, u f is the solution of minimal L 2 (D, dA α ) norm of the equation ∂v = u f ′ . Now, it is well known that the solution of ∂v = u f ′ given by Indeed, this estimate follows from the Cauchy-Schwarz inequality and the fact that for c > 0 and t > −1, the integral is comparable to (1 − |z| 2 ) −c (this is just a variant of Lemma B). Taking all of this into account, we obtain It follows easily from (5.4) that the operator u is bounded (or compact) if sup z∈D (1 − |z|)|u(z)| < ∞ (or if lim |z|→1 − (1 − |z|)|u(z)| = 0), and it is clear that these conditions are implied by the fact that u ∈ A p p−2 . Now, let {e n } be an orthonormal set in D α . Therefore, using (5.4), Hölder's inequality, (2.3) and (2.7), we obtain n u e n A different proof for the case p = 2 (which can be adapted to the case p > 2) is as follows. Let {e n } be an orthonormal basis of D α , and take 0 < ε < 1. Lemma 5.2 then yields | u e n (w)| 2 dA α (w) For 1 < p < 2, one has A p p−2 ⊂ A 2 . Thus, by the case we have just proved, the operator u is Hilbert-Schmidt and, in particular, compact. By Proposition 2.2, a sufficient condition for u to be in the class S p is Now, take 0 < ε < 1 with α − ε > −1 and p − εp > 1. Proceeding as in (5.5) and then using Lemma C, we obtain This, together with Lemma C, gives Thus, Now, consider an r-lattice {a n } with associated hyperbolic disks {D n }. Since Inserting this into (5.7) and applying Lemma B, we obtain Theorem 0 ]. This establishes (5.6).
Thus, putting this all together, we see that if g ∈ B p , then Observe that we have just proved one implication; but the other is proved exactly in the same way. Finally, it is clear that { f n } is an orthonormal subset ofḊ 2+α if and only if { f ′ n } is an orthonormal subset of A 2 2+α . It follows that (a) is equivalent to (b).
The next result shows that M g ′ ∈ S 1 is not equivalent to M g ′′ being in the trace class. We recall that g ∈ B 1 if g ∈ H (D) and D |g ′′ (z)|dA(z) < ∞. . Now, from this and (5.12), we see that the sufficient condition in part (c) is sharp in a certain sense. Also, since g a B 1 ≍ (1 − |a| 2 ) −γ , we see that part (b) can not be reversed.
To see that part (c) cannot be reversed, consider a lacunary series g(z) = and, therefore, the singular numbers λ n satisfy λ n ≍ 1/ (n + 1) 3 (n − j + 1) . Thus, M z j S 1 = n≥ j λ n ≍ n≥ j 1 (n + 1) 3 (n − j + 1) ≍ j −1 , and substituting this into (5.13) gives This, together with part (b), shows that given a lacunary series g(z) = k a k z n k , the multiplication operator M g ′′ : D → A 2 2 belongs to S 1 if and only if k n k |a k | < ∞. It is well known that this condition is equivalent to g being in B 1 [30, p. 100]. Now, given a function ϕ as described in part (d), it is straightforward to select numbers {a k } and the sequence {n k } such that the summability condition k n k |a k | < ∞ is met and D |g ′′ |ϕdA = ∞.

Acknowledgments.
The authors thank the referee for comments and suggestions that improved the final version of the paper and also for pointing out the relation between the integration operator T g and the previous work of A. Calderón.